Question

Suppose that there are two assets A and B. Their expected returns are E(rA) and E(rB), where E(rA) < E(rB), and the risk free-rate is rf = 0. Their variances, greater than zero, are sd(A)^2and sd(B)^2 and they are the same . Their covariance is zero. In this question assume that shorting assets is not possible. a. (2 points) What is the variance of the portfolio that puts weight w on asset A and 1-w on asset B? Simplify the expression so that sd(B)^2 does not appear. b. (2 points) What is the minimum variance portfolio? c. (1 point) What is the variance of the minimum variance portfolio? d. (2 points) What is the Sharpe ratio of the minimum variance portfolio? e. (2 points) What must be true for the Sharpe ratio of Asset B to be higher than that of the minimum variance portfolio? f. (3 points) Is it possible for the optimal risky portfolio to be the same as the minimum variance portfolio? Very briefly explain the intuition for your answer. g. (3 points) Does your result in part (e) of this question imply that there are no benefits to diversification when two assets are uncorrelated? h. (3 points) Explain the justification for constructing optimal complete portfolios using only the optimal risky portfolio and the risk-free asset.

Answer 1

a. Variance of asset A - sd(A)^2

Variance of asset B - sd(B)^2

(Note : Variance means the square of the Standard Deviation of an asset's returns so the Standard Deviation of A and B will be sd(A) and sd(B) )

Also given, that both variances are the same which means sd(A)^2 = sd(B)^2

Expected returns of Asset A and B are E(rA) and E(rB) respectively

Weights of A and B are w and 1-w respectively

Covariance between both assets is zero (0)

Variance of a portfolio consisting of 2 assets = [sd(A)² * w(A)² ] + [sd(B)² * w(B)² ] + [2 * w(A) * w(B) * Cov (A,B)]

where w(A)² and w(B)² denotes the square of the weights of asset A and B, Cov (A,B) denotes the covariance of the assets.

Hence, Variance of the given portfolio = [sd(A)² * w² ] + [sd(B)² * (1-w)² ] + [2 * w * 1-w * 0]

= [sd(A)² * w² ] + [sd(B)² * (1 + w² - 2w) ] (Note : the last expression gets eliminated because the covariance is zero)

= [sd(A)² * w² ] + [ sd(B)² + sd(B)² * w² - sd(B)² * 2w ]

Now since  sd(A)^2 = sd(B)^2 we convert all terms with sd(B) into sd(A)

= [sd(A)² * w² ] +  [sd(A)² ] + [sd(A)² * w² ] - [sd(A)² * 2w]

= 2 * sd(A)² * w²  - 2 * sd(A)² * w + sd(A)²

b. Minimum variance portfolio :

For this we need to find the weight of each asset A and B in the minimum variance portfolio as per the formula below:

Weight of Asset A in the minimum variance portfolio = [ sd(B)² - (Cov A,B) ] / [ sd(A)² + sd(B)² - 2 * (Cov A,B) ]

Since Cov(A,B) = 0 ,

Weight of Asset A = sd(B)² / sd(A)² + sd(B)²

(Given that sd(A)^2 = sd(B)^2 ) , therefore above expression equals:

sd(A)² / sd(A)² + sd(A)²

= sd(A)² / 2 sd(A)²

= 1 / 2

= 0.50

Therefore Weight of Asset B in the minimum variance portfolio = 1 - W(A) = 1 - 0.50 = 0.50

The minimum variance portfolio is the portfolio consisting of 50% in weight each of assets A and B.

Expected Return of the minimum variance portfolio = E(rA) * W(A) + E(rB) * W(B)

= 0.50 E(rA) + 0.50 E(rB)

c. Variance of the minimum variance portfolio = [ sd(A)² * 0.50² ] + [ sd(B)² * (1-0.50)² ] + [2 * 0.50 * 1-0.50 * 0]

= 0.25 sd(A)² + 0.25 sd(B)²

= 0.25 sd(A)² + 0.25 sd(A)²

= 0.50 sd(A)²

d. Sharpe Ratio = [ E(rP) - rf ] / sd(P)

where E(rP) denotes Expected Return of a Portfolio (which is calculated above in part b), rf denotes the risk-free rate which is given as 0, sd(P) denotes the Standard Deviation of the Portfolio (which is equal to square root of the variance - calculated in part c above)

Hence, Sharpe Ratio of the minimum variance portfolio​​​​​​​ = [ 0.50 E(rA) + 0.50 E(rB) ] - 0 / square root (0.50 sd(A)² )

= 0.50 [E(rA) + E(rB)] / 0.70716 sd(A)